Interferometric optical techniques are widely used to measure surface profiles of precision optical components.
For example, to measure the surface profile of a test surface, one can use an interferometer to combine a test wavefront reflected from the test surface with a reference wavefront reflected from a reference surface to form an optical interference pattern. Spatial variations in the intensity profile of the optical interference pattern correspond to phase differences between the combined test and reference wavefronts caused by variations in the profile of the test surface relative to the reference surface. Phase-shifting interferometry (PSI) can be used to accurately determine the phase differences and the corresponding profile of the test surface. However, the surface profile measurement of the test surface is only relative to the surface profile of the reference surface, which is assumed to be perfect (e.g., flat) within the tolerances of the measurement.
With PSI, the optical interference pattern is recorded for each of multiple phase-shifts between the reference and test wavefronts to produce a series of optical interference patterns that span, for example, at least a full cycle of optical interference (e.g., from constructive, to destructive, and back to constructive interference). The optical interference patterns define a series of intensity values for each spatial location of the pattern, wherein each series of intensity values has a sinusoidal dependence on the phase-shifts with a phase-offset equal to the phase difference between the combined test and reference wavefronts for that spatial location. Using numerical techniques known in the art, the phase-offset for each spatial location is extracted from the sinusoidal dependence of the intensity values to provide a profile of the test surface relative the reference surface. Such numerical techniques are generally referred to as phase-shifting algorithms.
The phase-shifts in PSI can be produced by changing the optical path length from the measurement surface to the interferometer relative to the optical path length from the reference surface to the interferometer. For example, the reference surface can be moved relative to the measurement surface. Alternatively, the phase-shifts can be introduced for a constant, non-zero optical path difference by changing the wavelength of the measurement and reference wavefronts. The latter application is known as wavelength tuning PSI and is described, e.g., in U.S. Pat. No. 4,594,003 to G. E. Sommargren.
For many applications (e.g., certification of optical components, calibrating a transfer standard, etc.), one desires an absolute measurement of the surface profile of test surface, i.e., a surface profile measurement that is independent of the reference surface used in the interferometric measurement. This is called absolute figure metrology.
Absolute figure metrology for the certification of a flat surface has been a long-standing problem in optical metrology. A summary of proposed solutions can be found in, for example, D. Malacara, Optical Shop Testing, 2nd Ed., Chap 14, John Wiley & Sons, New York (1992). For flats, prior-art methods rely, at least in part, on the relative measurements of 3 unknown surfaces with respect to each other—the so-called 3-flat test—from which one can solve for the absolute shape of these surfaces along a single meridian. Information about the absolute form of the remainder of the surfaces is obtained with additional rotations to determine other meridians and by invoking symmetry requirements or by a combination of additional rotations, symmetry and/or polynomial fitting to fill in the missing gaps. Similar arguments hold for spheres.
Such techniques typically require enormous effort. Many manipulations of the parts under test are required, requiring both precise yet flexible motion control. Furthermore, the manual nature of the measurement does not lend itself easily to error estimation and makes human error a major contributor to the ultimate uncertainty in the results. Finally, the difficulty involved in producing certification to nanometer levels makes these measurements only possible with highly skilled personnel, making the cost of these measurements prohibitive.